set theory

Set theory is a branch of mathematics created by Georg Cantor at the end of the nineteenth century. Initially controversial, set theory has come to play the role of a foundational theory in modern mathematics, in the sense of a theory invoked to justify assumptions made in mathematics concerning the existence of mathematical objects (such as numbers or functions) and their properties. Formal versions of set theory also have a foundational role to play as specifying a theoretical ideal of mathematical rigor in proofs. Cantor's basic discovery was that if we define two sets A and B to have the same number of members (the same cardinality) then there is a way of pairing off members of A exhaustively with members of B. The appearance around the turn of the century of the so-called set-theoretical paradoxes, such as Russell's paradox, prompted the formulation in 1908 by Ernst Zermelo of an axiomatic theory of sets. The axioms for set theory now most often studied and used are those called the Zermelo-Fraenkel axioms, usually together with the axiom of choice. The Zermelo-Fraenkel axioms are commonly abbreviated to ZF, or ZFC if the axiom of choice is included. An important feature of ZFC is that every object that it deals with is a set. In particular, every element of a set is itself a set. Other familiar mathematical objects, such as numbers, must be subsequently defined in terms of sets.


Some basic set theory

A set is a collection of objects or quantities symbolized by a capital letter. Thus


S = {2, 4, 6, 8}


means that S is the set of even numbers less than 10. A member of a set is called an element: symbolically, 2 ∈ S means that 2 is an element of S; 3 ∉ means that 3 is not an element of S.


An order set is equivalent to a sequence. A set may be infinite, finite, or empty: the set of all even numbers is infinite, that of all those less than 10 is finite, and that of all those less than 1 is empty. This empty set, or null set, symbolized by {}, should not be confused with {0}, which is a set with one member, zero.


If there is a one-to-one correspondence between the elements of two sets, then they are said to be equivalent, and if the sets have identical elements they are equal:


S1 = {a, b, c, d}
S2 = {e, f, g, h}
S3 = {d, c, b, a} shows three equivalent sets. Moreover, S1 = S2. Two equivalent sets are writter S1S2.


If some elements of one set are also elements of another, then those elements are called the intersection of the two sets, symbolized S1S2. The sets of all elements that are members of at least one of the two sets is their union, written S1S2. A set whose members are all members of another set is termed a subset. Thus, if


S1 = {a, b, c, d, e}
S2 = {b, d, f, g}
then S1S2 = {b, d} and S1S2 {a, b, c, d, e, f, g}


Moreover, S1 and S2 are subsets of S1S2, written S1S1S2 and SsS1S2. The set of all elements of all the sets in a particular discussion is the universal set, or domain, symbolized by U. The domain may contain elements in addition to all those under discussion.


Set theory is of importance throughout mathematics. In analytical geometry, for example, a curve may be considered as a set of points, or point set. For two functions, f(x) and g(x), represented by the sets Sf and Sg, SfSg gives those points at which the curves intersect (see intersection). Sets can be represented pictorially by Venn diagrams.